How many integer values of $n$ satisfy $-50 < n^3 < 50$?
Answer: We are asked to find the integers whose cubes are between $-50$ and $50$.  Since $f(x)=x^3$ is a monotonically increasing function, we can find the least and the greatest integers satisfying the inequality and count the integers between them, inclusive (see graph).  Since $3^3=27<50$ and $4^3=64>50$, $n=3$ is the largest solution.  Similarly, $n=-3$ is the smallest solution.  Therefore, there are $3-(-3)+1=\boxed{7}$ solutions.   [asy]size(7cm,8cm,IgnoreAspect);
defaultpen(linewidth(0.7));
import graph;

real f(real x)
{

return x*x*x;
}

draw(graph(f,-4.5,4.5),Arrows(4));

draw((-4.5,50)--(4.5,50),linetype("3 4"),Arrows(4));
draw((-4.5,-50)--(4.5,-50),linetype("3 4"),Arrows(4));

xaxis(-4.5,4.5,Arrows(4));
yaxis(-4.5^3,4.5^3,Arrows(4));

label("$y=50$",(6,50));
label("$y=-50$",(6,-50));
label("$x$",(4.5,0),E);
label("$f(x)=x^3$",(0,4.5^3),N);

int n;
for(n=-3;n<=3;++n)

{

dot((n,n^3));

}
dot((-4,-64),NoFill);
dot((4,64),NoFill);
label("$(3,27)$",(3,27),W);
label("$(4,64)$",(4,64),W);[/asy]